Consider the limit L= _ x 0^ + x^ x

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the limit L=lim⁡x→0+xsin⁡x.L=\lim_{x\to 0^{+}}x^{\sin x}. Although the numerical value of LL is ultimately found by a direct computation, insight into the algebraic structure of xsin⁡xx^{\sin x} is required before the limit can be evaluated. (a) Identify the indeterminate form that arises when the expression xsin⁡xx^{\sin x} is written in a form suitable for limit evaluation as x→0+x\to 0^{+}. Briefly justify your answer. (b) Explain, without completing the full calculation, why introducing the natural logarithm of xsin⁡xx^{\sin x} converts the indeterminate form you identified in part (a) into a ratio of two quantities whose limit can be analyzed with standard limit techniques. Your explanation should reference the structural change that occurs after the logarithm is applied.

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